Unit 1: Convexity Mathematics II Departament de Matemtiques per a - PowerPoint PPT Presentation

Convex sets Convex and concave functions Unit 1: Convexity Mathematics II Departament de Matemàtiques per a l’Economia i l’Empresa Academic year 2010/2011 Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Index Convex sets 1 Convex and concave functions 2 Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Index Convex sets 1 Convex and concave functions 2 Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Teorema Theorem If A is a diagonal matrix, then: it is positive definite if all the entries in the main diagonal are > 0 . it is negative definite if all the entries in the main diagonal are < 0 . positive semi-definite if all the entries in the main diagonal are ≥ 0 . negative semi-definite if all the entries in the main diagonal are ≤ 0 indefinite if it contains entries > 0 y < 0 in the main diagonal. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definitions Definition A square matrix is regular if its determinant is � = 0 . Definition A square matrix is singular if its determinant is 0. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definitions Definition The k-th order principal minors of a n × n matrix A are the determinants of the submatrices formed by k rows of A (in order) and the same k columns. Definition The k-th order leading principal minor of a n × n matrix A is the principal minor formed by the k first rows and the k first columns of A. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0 , then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0 , i.e., A 1 < 0 , A 2 > 0 , . . . , then A is negative definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0 , then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0 , i.e., A 1 < 0 , A 2 > 0 , . . . , then A is negative definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0 , then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0 , i.e., A 1 < 0 , A 2 > 0 , . . . , then A is negative definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method Theorem Let A be a n × n regular square matrix. If all the leading principal minors are > 0 , then A is positive definite. If the leading principal minors of odd order are < 0 and those of even order are > 0 , i.e., A 1 < 0 , A 2 > 0 , . . . , then A is negative definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method (2) Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0 , then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0 , i.e., A 1 ≤ 0 , A 2 ≥ 0 , . . . , then A is negative semi-definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method (2) Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0 , then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0 , i.e., A 1 ≤ 0 , A 2 ≥ 0 , . . . , then A is negative semi-definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method (2) Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0 , then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0 , i.e., A 1 ≤ 0 , A 2 ≥ 0 , . . . , then A is negative semi-definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Jacobi method (2) Theorem Let A be a n × n singular square matrix. If all the principal minors are ≥ 0 , then A is positive semi-definite. If the principal minors of odd order are ≤ 0 and those of even order are ≥ 0 , i.e., A 1 ≤ 0 , A 2 ≥ 0 , . . . , then A is negative semi-definite. In any other case, A is indefinite. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Concepts Definition If p y q are two points in R n , p � = q, the line that goes through both points can be expressed as the set of points satisfying: x = ( 1 − λ ) p + λ q , λ ∈ R If we consider the points that satisfy: x = ( 1 − λ ) p + λ q , λ ∈ [ 0 , 1 ] then we get the points in the segment joining p and q . Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Concepts Definition If p y q are two points in R n , p � = q, the line that goes through both points can be expressed as the set of points satisfying: x = ( 1 − λ ) p + λ q , λ ∈ R If we consider the points that satisfy: x = ( 1 − λ ) p + λ q , λ ∈ [ 0 , 1 ] then we get the points in the segment joining p and q . Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definition Definition A set C ⊂ R n is convex if for any pair of points x , y ∈ C and 0 ≤ λ ≤ 1 , then ( 1 − λ ) x + λ y ∈ C. This means that C is convex if for any pair of points in C , the points in the segment that joins them are in C , too. The empty set ∅ and sets containing only one point are convex. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definition Definition A set C ⊂ R n is convex if for any pair of points x , y ∈ C and 0 ≤ λ ≤ 1 , then ( 1 − λ ) x + λ y ∈ C. This means that C is convex if for any pair of points in C , the points in the segment that joins them are in C , too. The empty set ∅ and sets containing only one point are convex. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definition Definition A set C ⊂ R n is convex if for any pair of points x , y ∈ C and 0 ≤ λ ≤ 1 , then ( 1 − λ ) x + λ y ∈ C. This means that C is convex if for any pair of points in C , the points in the segment that joins them are in C , too. The empty set ∅ and sets containing only one point are convex. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Hyperplane Definition An hyperplane in R n is a subset of the form H = { x ∈ R n | c 1 x 1 + c 2 x 2 + . . . + c n x n = α } where c i , α ∈ R . If we denote by c the vector whose elements are c i , we can express the previous set as H = { x ∈ R n | cx = α } . Theorem Hyperplanes are convex sets. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Semispace Definition A semispace in R n is a subset of the form S = { x ∈ R n | c 1 x 1 + c 2 x 2 + . . . + c n x n ≤ α } where c i , α ∈ R . Theorem Semispaces are convex sets. Theorem The intersection of convex sets is a convex set. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Semispace Definition A semispace in R n is a subset of the form S = { x ∈ R n | c 1 x 1 + c 2 x 2 + . . . + c n x n ≤ α } where c i , α ∈ R . Theorem Semispaces are convex sets. Theorem The intersection of convex sets is a convex set. Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definition Definition Let f : D ⊂ R n → R be a function defined on a convex set D. Then f is convex if for every pair x , y ∈ D and 0 ≤ λ ≤ 1 it holds: f (( 1 − λ ) x + λ y ) ≤ ( 1 − λ ) f ( x ) + λ f ( y ) Definition Let f : D ⊂ R n → R be a function defined on a convex set D. Then f is concave if for every pair x , y ∈ D and 0 ≤ λ ≤ 1 it holds: f (( 1 − λ ) x + λ y ) ≥ ( 1 − λ ) f ( x ) + λ f ( y ) Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Definition Definition Let f : D ⊂ R n → R be a function defined on a convex set D. Then f is convex if for every pair x , y ∈ D and 0 ≤ λ ≤ 1 it holds: f (( 1 − λ ) x + λ y ) ≤ ( 1 − λ ) f ( x ) + λ f ( y ) Definition Let f : D ⊂ R n → R be a function defined on a convex set D. Then f is concave if for every pair x , y ∈ D and 0 ≤ λ ≤ 1 it holds: f (( 1 − λ ) x + λ y ) ≥ ( 1 − λ ) f ( x ) + λ f ( y ) Mathematics II Unit 1: Convexity

Convex sets Convex and concave functions Theorem Theorem Let f : D ⊂ R n → R be a class C 2 function on a convex open set D. If Hf ( x ) is positive definite (or semi-definite) for every point 1 x ∈ D, f is convex. If Hf ( x ) is negative definite (or semi-definite) for every 2 point x ∈ D, f is concave. Mathematics II Unit 1: Convexity